The variational method in contact problems. The present state of the problem and trends in its development

Research on variational methods for solving problems on the contact of solid deformable bodies is reviewed and trends in the development of these researches at the present time are analysed. The Signorini problem and it generalizations, numerical methods, different models of friction, investigations into the problem of the existence and uniqueness of a solution, the problem of rolling motion, the problem of describing the boundary conditions, inelastic materials and problems of contact dynamics and electro-elastic contact are considered. The analysis shows that research on the problem of the contact of deformable bodies is being conducted over a broad front in different areas and the results are being applied in different areas of modern engineering and technology.     

The axisymmetric contact problem taking into account transient heat production due to sliding friction

The contact problem of the sliding of a solid heat insulator with a plane surface along the boundary of an axisymmetric elastic body is considered, taking into account heat release and the thermal distortion of the boundary of the deformable body due to friction. It is assumed that the shear stresses have no effect on the value of the contact pressures, which enables the problem to be investigated in an axisymmetric formulation. The solution is constructed in two stages: first the form of the thermally distorted surface is determined using known expressions, obtained by Carslaw and Jaeger and also by Barber, and then the contact condition is considered taking into account the elastic displacements and distortion of the form of the surface due to heating, and the integral equation of the problem for determining the unknown contact pressures is derived. The latter equation is solved numerically by approximating the unknown contact pressures by a piecewise-constant function.     

Pressure of a plane stamp of nearly circular cross section on an elastic half-space : PMM vol. 40, n 6, 1976, pp. 1143–1145

An approximate method of solving the contact problem of impressing a plane stamp of nearly circular cross section into an elastic half-space is suggested. The friction of the contact surface is neglected. A numerical algorithm for the method is produced. An elliptical and rectangular stamps are considered as examples.There is no general method of solving the problems for stamps of nearly circular cross section. Apart from the classical problem of a plane elliptical stamp, the literature gives solutions for the problems of polygonal stamps, with each problem however requiring a different approach. An approximate solution for the problem of impressing a stamp of nearly circular cross section into an elastic half-space is given in [1]. The method makes it possible to use the same approach to solve the contact problem for an arbitrary region of contact, and to construct an universal numerical algorithm. The program can be adapted to each particular case by making the corresponding changes in the procedure of computing the Fourier coefficients of the equation of the boundary of the area of contact. Below a numerical algorithm for the approximate method in question is given. A more effective formulation of the solution is given for the case of the elliptical stamp.     

Meshless solution of two-dimensional incompressible flow problems using the radial basis integral equation method

The two-dimensional incompressible fluid flow problems governed by the velocity–vorticity formulation of the Navier–Stokes equations were solved using the radial basis integral (RBIE) equation method. The RBIE is a meshless method based on the multi-domain boundary element method with overlapping subdomains. It solves at each node for the potential and its spatial derivatives. This feature of the RBIE is advantageous in solving the velocity–vorticity formulation of the Navier–Stokes equations since the calculated velocity gradients can be used to compute the vorticity that is prescribed as a boundary condition to the vorticity transport equation. The accuracy of the numerical solution was examined by solving the test problem with known analytical solution. Two benchmark problems, i.e. the lid driven cavity flow and the thermally driven cavity flow were also solved. The numerical results obtained using the RBIE showed very good agreement with the benchmark solutions.     

Modeling incompressible flows using a finite particle method

This paper describes the applications of a finite particle method (FPM) to modeling incompressible flow problems. FPM is a meshfree particle method in which the approximation of a field variable and its derivatives can be simultaneously obtained through solving a pointwise matrix equation. A set of basis functions is employed to obtain the coefficient matrix through a sequence of transformations. The finite particle method can be used to discretize the Navier–Stokes equation that governs fluid flows. The incompressible flows are modeled as slightly compressible via specially selected equations of state. Four numerical examples including the classic Poiseuille flow, Couette flow, shear driven cavity and a dam collapsing problem are presented with comparisons to other sources. The numerical examples demonstrate that FPM is a very attractive alternative for simulating incompressible flows, especially those with free surfaces, moving interfaces or deformable boundaries.     

The contact problem with the bulk application of intermolecular interaction forces (a refined formulation)

A refined formulation of the contact problem when there are intermolecular interaction forces between the contacting bodies is considered. Unlike the traditional formulation, it is assumed that these forces are applied to points within the body, rather than to the surface of the deformable body as a contact pressure, and that the body surface is load-free. Solutions of the contact problems for a thin elastic layer attached to an absolutely rigid substrate and for an elastic half-space are analysed. The refined and traditional formulations of the problem when there is intermolecular interaction are compared. ©2013     

On the algorithm of the solution of the signorini problem

An algorithm is proposed for solving the Signorini problem /1/ in the formulation of a unilateral variational problem for the boundary functional in the zone of possible contact /2/. The algorithm is based on a dual formulation of Lagrange maximin problems for whose solution a decomposition approach is used in the following sense: a Ritz process in the basis functions that satisfy the linear constraint of the problem, the differential equation in the domain, is used in solving the minimum problem (with fixed Lagrange multipliers); the maximum problem is solved by the method of descent (a generalization of the Frank-Wolf method) under convexity constraints on the Lagrange multipliers. The algorithm constructed can be conisidered as a modification of the well-known algorithm to find the Udzawa-Arrow-Hurwitz saddle points /3, 4/. The convergence of the algorithm is investigated. A numerical analysis of the algorithm is performed in the example of a classical contact problem about the insertion of a stamp in an elastic half-plane under approximation of the contact boundary by isoparametric boundary elements. The comparative efficiency of the algorithm is associated with the reduction in the dimensionality of the boundary value problem being solved and the possibility of utilizing the calculation apparatus of the method of boundary elements to realize the solution.     

The solution of contact problems using boundary element method

The formulation of contact problems is extended to the case of moving punches and to the case when the state of the systems being investigated depends on the history of the change in the external actions. The quasi-static contact problem for a moving rigid rough punch and a single linearly deformable body is considered. A new iterational process is proposed for solving contact problems, taking friction in the contact area into account, and its convergence is proved. An algorithm of the solution, based on the boundary element method, is developed. Solutions of specific problems are given and analysed. Estimates of the difference of the solutions due to the difference in the impenetrability conditions and the difference in the steps of the loading parameter are obtained.     

A partitioning algorithm for the network loading problem

This paper proposes a Benders-like partitioning algorithm to solve the network loading problem. The approach is an iterative method in which the integer programming solver is not used to produce the best integer point in the polyhedral relaxation of the set of feasible capacities. Rather, it selects an integer solution that is closest to the best known integer solution. Contrary to previous approaches, the method does not exploit the original mixed integer programming formulation of the problem. The effort of computing integer solutions is entirely left to a pure integer programming solver while valid inequalities are generated by solving standard nonlinear multicommodity flow problems. The method is compared to alternative approaches proposed in the literature and appears to be efficient for computing good upper bounds.     

An approach to solving axially symmetric problems for spherical orthotropic bodies

A series expansion method is developed in which the small parameter is the deviation of the spherically orthotropic properties of deformable bodies from their transversally isotropic properties. The problem is reduced to a rigorous analytic solution of inhomogeneous boundary value problems. The efficiency of the approximation technique developed here and its practical convergence are examined in a centrally symmetric problem for an orthotropic sphere which permits an exact analytic solution. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 17–24, 1999.     

An ultra-weak method for acoustic fluid–solid interaction

We introduce the ultra-weak variational formulation (UWVF) for fluid–solid vibration problems. In particular, we consider the scattering of time-harmonic acoustic pressure waves from solid, elastic objects. The problem is modeled using a coupled system of the Helmholtz and Navier equations. The transmission conditions on the fluid–solid interface are represented in an impedance-type form after which we can employ the well known ultra-weak formulations for the Helmholtz and Navier equations. The UWVF approximation for both equations is computed using a superposition of propagating plane waves. A condition number based criterion is used to define the plane wave basis dimension for each element. As a model problem we investigate the scattering of sound from an infinite elastic cylinder immersed in a fluid. A comparison of the UWVF approximation with the analytical solution shows that the method provides a means for solving wave problems on relatively coarse meshes. However, particular care is needed when the method is used for problems at frequencies near the resonance frequencies of the fluid–solid system.     

The solution of three-dimensional friction contact problems

A variational method is developed for solving friction contact problems, in which the friction obeys Coulomb's of friction law in velocities, and numerical solutions of three-dimensional problems of the contact of a sphere, a cylinder of finite length and a cube with an elastic half-space are constructed. It is established that the maximum frictional forces correspond to a boundary point of the regions of adhesion and slippage. When the number of steps,increase this maximum decreases, and the distribution of the frictional forces becomes smoother. Certain undesirable effects that can arise during numerical implementation of the method – numerical artefacts – are described. These effects can occur in the numerical solution of problems with a different physical content, the mathematical structure of which is similar to the structure of the contact problems investigated, as the artefacts are caused by the presence of unilateral constraints and by the dependence on external effects of the region in which unilateral constraints with an equally sign occur. This problem is solved by an appropriate choice of the load-step zero approximations.     

A New Non-Overlapping Domain Decomposition Method for a 3D Laplace Exterior Problem

We propose a method for solving three-dimensional boundary value problems for Laplace’s equation in an unbounded domain. It is based on non-overlapping decomposition of the exterior domain into two subdomains so that the initial problem is reduced to two subproblems, namely, exterior and interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To match the solutions on the interface between the subdomains (the sphere), we introduce a special operator equation approximated by a system of linear algebraic equations. This system is solved by iterative methods in Krylov subspaces. The performance of the method is illustrated by solving model problems.     

Micromechanical Analysis of Inelastic Deformation of Unidirectional Fibrous Composites under Multiaxial and Shear Loadings

An algorithm is proposed for numerically solving nonlinear 3D problems of micromechanics of a unidirectionally reinforced composite with a regular structure. For the matrix, equations of the deformation theory of plasticity and relations of reduced rigidity in its failure zones are used, whereas the fibers are elastic and indestructible. According to the method of local approximation, fields of microstresses and microstrains are determined in a structural fragment containing nine periodic cells. Boundary conditions of the fragment correspond to an arbitrary combination of longitudinal, transverse, and shear microstresses occurring in the central part of the fragment. The solution to the nonlinear 3D problem is sought by the method of superposition with an iterational refinement based on the successive solution of an antiplane problem and a problem on a generalized plain strain state of the structural segment. Special features of the iteration procedure are considered. The calculated deformation diagrams and ultimate strengths of a unidirectional glass-epoxy composite are presented for several loading trajectories.     

The interpolation solution of the second basic plane problem of the dynamics of elastic solids

The second basic plane problem of the dynamics of elastic bodies is considered in the Muskhelishvili formulation, when the known boundary displacements are replaced by interpolation time polynomials and the known initial conditions are replaced by polyharmonic functions, which interpolate the initial conditions in a region with a finite number of interpolation nodes. In this case a solution of the problem, called here the interpolation solution, is possible. It must satisfy the dynamic equations and interpolate the boundary displacements and initial displacements and velocities. This solution is constructed in the form of a polynomial and is reduced to solving a series of boundary-value problems for determining the coefficients of this polynomial.     

Optimal compromise solution of multi-objective minimal cost flow problems in fuzzy environment

Ghatee and Hashemi [M. Ghatee, S.M. Hashemi, Ranking function-based solutions of fully fuzzified minimal cost flow problem, Inform. Sci. 177 (2007) 4271–4294] transformed the fuzzy linear programming formulation of fully fuzzy minimal cost flow (FFMCF) problems into crisp linear programming formulation and used it to find the fuzzy optimal solution of balanced FFMCF problems. In this paper, it is pointed out that the method for transforming the fuzzy linear programming formulation into crisp linear programming formulation, used by Ghatee and Hashemi, is not appropriate and a new method is proposed to find the fuzzy optimal solution of multi-objective FFMCF problems. The proposed method can also be used to find the fuzzy optimal solution of single-objective FFMCF problems. To show the application of proposed method in real life problems an existing real life FFMCF problem is solved.     

Nonlinear statics and dynamics of antisymmetric composite laminated square plates supported on nonlinear elastic subgrade

Non-linear static and dynamic analysis is presented for composite laminated anti-symmetric square plates supported on non-linear elastic foundation subjected to uniformly distributed transverse and step loading, respectively. The formulation is based on first order shear deformation theory (FSDT) and Von-Karman non-linearity, subgrade interaction is modeled as shear deformable with cubic nonlinearity. The methodology of solution is based on the Chebyshev series technique. The coupled non-linear partial differential equations are linearized using quadratic extrapolation technique. Houbolt time marching scheme is employed for temporal discretisation. An incremental iterative approach is employed for the solution. The effects of foundation stiffness parameters and boundary conditions on the non-linear static and dynamic analysis on the central response are studied.     

Outcome-Space Cutting-Plane Algorithm for Linear Multiplicative Programming

This article presents an outcome-space pure cutting-plane algorithm for globally solving the linear multiplicative programming problem. The framework of the algorithm is taken from a pure cutting-plane decision set-based method developed by Horst and Tuy for solving concave minimization problems. By adapting this method to an outcome-space reformulation of the linear multiplicative programming problem, rather than applying directly the method to the original decision-set formulation, it is expected that considerable computational savings can be obtained. Also, we show how additional computational benefits might be obtained by implementing the new algorithm appropriately. To illustrate the new algorithm, we apply it to the solution of a sample problem.     

The current state of the problem of interaction of acoustic beams with obstacles in a deformable medium

We analyze the state of the problem of the formation of radiated and scattered acoustic beams in application to the development of a methodology for studying the information aspects of hydroacoustics and nondestructive control. We discuss the problems of the selective generation of characteristic vibrations of elastic objects in a deformable medium using sharply directed acoustic impulses. We study the problem of posing and methods of solving a certain class of inverse problems of scattering theory.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 56–64.In conclusion we note the papers [7, 74, 100] connected with the traditional method of solving inverse problems-the selection method.     

Multicommodity network flows: The impact of formulation on decomposition

This paper investigates the impact of problem formulation on Dantzig—Wolfe decomposition for the multicommodity network flow problem. These problems are formulated in three ways: origin-destination specific, destination specific, and product specific. The path-based origin-destination specific formulation is equivalent to the tree-based destination specific formulation by a simple transformation. Supersupply and superdemand nodes are appended to the tree-based product specific formulation to create an equivalent path-based product specific formulation. We show that solving the path-based problem formulations by decomposition results in substantially fewer master problem iterations and lower CPU times than by using decomposition on the equivalent tree-based formulations. Computational results on a series of multicommodity network flow problems are presented.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.